It is absolutely not rigorous enough, I'm afraid. Sure, it feels wasteful to cover any part of the disc twice. But it also feels wasteful to cover bits near th

It is absolutely not rigorous enough, I'm afraid. Sure, it feels wasteful to cover any part of the disc twice. But it also feels wasteful to cover bits near the edge with strips that have only a short length of overlap with the circle. It's not obvious that there isn't some clever way to reduce the second kind of waste that requires you to commit the first kind. Perhaps the following observation will help. Replace the circle with a "+" shape made out of five equal squares (like the cross on the Swiss flag). The most efficient way to cover this with strips is (I think -- I haven't actually tried to prove it) to use two strips "along" the arms of the cross. Those overlap in the middle, but they still do the job more efficiently than using (say) a single strip of 3x the width. So, how do you know that nothing like that happens with a disc instead of a cross?